Ultracold atoms in optical lattices have been demonstrated to be very promising for studying quantum many-body physics.^[1] In particular, quantum gas microscopy (QGM), composed of optical lattices and high-resolution imaging approaches, has becoming a cutting edge method for studying quantum crystals in a brand-new way.^[2–7] Experiments such as measurement of Rényi entropy of a few-body system,^[8] and in situ observation of many-body localization^[9] have been performed with QGM, making it a unique tool for quantum simulation with ultracold atoms.

In three-dimensional (3D) lattices, it is of challenge to distinguish atoms in neighboring layers due to the fact that the depth of focus of the microscope is usually larger than the lattice constant. Moreover, the fluorescence emitted by out-of-focus atoms will blur the signal arising from the in-focus plane, damaging the imaging contrast. Therefore, it is important to prepare atoms in a monolayer for further studies. Furthermore, the 2D system is of great research interest for observing various quantum phenomena such as the Berezinsky–Kosterlitz–Thouless (BKT) transition.^[10,11]

In principle, there are two different approaches to prepare 2D quantum gases. One is to compress a three-dimensional cloud of ultracold atoms with light sheets and load them into one layer of a one-dimensional (1D) optical lattices,^[7,12] or into a surface trap.^[13,14] The other approach, as demonstrated herein, starts with an 1D optical lattice consisting parallel pancake-like traps filled with atoms. Then a microwave “knife” is used to flip the internal state of one layer of the Zeeman-shifted atoms in the presence of a gradient magnetic field. The rest atoms which are not selected will be blown away from the system by a resonant laser beam.^{[3,4,15,16,6]}

In order to perform future experiments with a prepared 2D system, repeatedly producing the 2D system with high stability is of key importance. This would ensure the high-fidelity manipulation of many-body quantum state and a sufficient signal-to-noise ratio (SNR) during imaging. A low fluctuation of the number of atoms favors the evaporative cooling to achieve lower temperatures for studying quantum phenomena. For example, a low temperature is critical to create unity filling Mott insulators^[17,18] and to observe antiferromagnetic correlations^[19,20] in optical lattices.

With this approach, the major challenges for preparing a 2D quantum gas are: Firstly, to provide a gradient magnetic field with low noises and slow drifts; Secondly, to compose a microwave pulse for coupling relevant atomic levels with high fidelity. Here we report on the techniques we developed to meet these demands. Briefly, based on the measured signal with a Hall sensor the current noise of the power supply is suppressed by a fast feedback circuit and the current drift is corrected. The environmental magnetic field is monitored by a fluxgate type sensor and its drift is further compensated with a pair of Helmholtz coils whose current is delivered by an ultra-stable power supply. A microwave Landau–Zener transfer process is used to flip the internal state of atoms. The achieved efficiency, 99.92(3)%, of flipping atomic states indicates the performance of noise suppression. A repeated measurement on the spatial drift of the flipped atoms calibrates the drift of the magnetic field as 0.042(3) mG/hour, which outperforms the best results reported in a couple similar experiments.^[16,21] In the following, we describe the experimental setup in Section 2. The overview on preparing the 2D quantum system is present in Section 3. The scheme on the applied magnetic field is given in Section 4. The methods on selecting one layer of the atoms by magnetic resonance are described in Section 5. The study on removing the atoms in other layers by a resonant laser beam is in Section 6. Noise reduction of the current supply is discussed in Section 7. Calibration of the method with microscopic imaging is provided in Section 8. Finally a short conclusion is presented in Section 9.

Our experimental setup is shown in Fig. 1. The pressure of the vacuum chamber (the volume in between the two glass viewports) is measured to be lower than 1.0 × 10⁻¹¹ mbar, maintained by a non-evaporable getter pump. A Bose–Einstein condensate (BEC) of 4.0(3) × 10^{6 87}Rb atoms in the state of |F = 1, m_F = −1 ⟩ is prepared in the center of the vacuum chamber via evaporative cooling in an cross dipole trap.

Fig. 1.

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	Fig. 1. Experimental setup. Parts of the compensation coils are shown, while the vacuum chamber made of stainless steel and the metal parts of the viewports are hidden. Each of X and Y lattices is formed by two laser beams of 532 nm crossed at an angle of 50°.

Fig. 2.

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	Fig. 2. Current supplies and the coils for providing a stabilized low-noise magnetic field. The commercial proportional–integral–derivative (PID) units (SRS, SIM960) control the output of BCS current sources to actively compensate the current drift of coils, while its output offset is modified every experimental trial to correct environmental magnetic field drift.

Beneath the chamber, a high-resolution objective situates with a numerical aperture of 0.8. The imaging resolution was calibrated to be 690 nm for the emitted light with a wavelength of 780 nm (⁸⁷Rb DII line). Meanwhile, the inner surface of the this viewport is coated with a high-reflective (HR) coating for the wavelength of 1064 nm. The incident laser beam of 1064 nm and its reflected beam from the inner surface together form the Z-lattice whose period is 532 nm. The X and Y lattices are both built by a pair of 532 nm laser beams crossed at the center with an angle of 50° resulting in a lattice constant of 630 nm. During fluorescence imaging, all the three lattices are raised to the maximal depth to pin the atoms on their own lattice sites.

After the BEC is achieved, the laser intensity of the Z-lattice is ramped up gradually and the atoms are loaded into a few layers of this lattice. Monolayer selection is performed with a Landau–Zener crossing (LZC) process,^[22] which is implemented by a microwave knife. The gradient magnetic field is generated by two pairs of coils mounted outside the chamber. The microwave field is delivered by an antenna placed on top of the upper viewport for approaching the atoms as close as possible.

A single-axis fluxgate (FL-200, StefanMayer Instrument) with a bandwidth of 1 kHz and a range of ± 2 G is placed about 3 cm away from the atoms to monitor the magnetic field along the Z direction. By adding three pairs of Helmholtz coils surrounding the chamber, we can generate bias fields along each direction to compensate for the stray magnetic field and adjust the direction of magnetic field gradient. The process will be described in detail in the following sections.

As mentioned above, the BEC atoms are loaded into a few layers of the Z-lattice by ramping up its trapping depth to k_B × 3.2 μ K corresponding to an axial trapping frequency of 2π × 24 kHz. We use two states in the ground state 5²S_1/2 for the monolayer selection, i.e. |F = 1,m_F = −1⟩ and |F = 2, m_F = −2⟩, labeled by |↓⟩ and |↑⟩, respectively. Therefore, we also refer to the process of transferring between |↓⟩ and |↑⟩ as “spin flip”. The gradient magnetic field is applied along the Z direction. Therefore, the Zeeman energy splitting of the atoms becomes spatially dependent.

To select one layer of the atoms in the Z-lattice, we take the following steps sequentially. First, all the atoms originally prepared in |↓⟩ are transferred to |↑⟩ with a global microwave sweep, an LZC pulse explained later in Section 5. Then, once all the atoms are in |↑⟩, a single layer of atoms are transferred back to |↓⟩ by a second LZC pulse with a narrower sweeping range. Finally, with a laser beam resonantly coupling to the transition |5²S_1/2,F = 2⟩ → |5² P_3/2, F = 3⟩, we push all the atoms in |↑⟩ out and leave this single layer of atoms in |↓⟩. To achieve a high precision and stability of these operations, several technical issues will be addressed in the future sections.

To perform spin flip for the atoms in one particular layer while leaving the rest unchanged, it is necessary to create spatially dependent energy splitting along the Z axis with a gradient magnetic field. Two points should be considered carefully. One is that the magnetic field gradient should be strong enough to make the energy splitting of atoms in neighboring layers highly distinguished by the second microwave pulse. The other one is that the magnetic field in the target layer should be homogeneous for a high-fidelity spin-flip of all the atoms. Write the gradient as

In total, we generate a magnetic field gradient of 100 G/cm combined with an offset magnetic field of 35 G at Z direction. Ideally, atoms should be prepared close to the center of the gradient coils at which gradient coils create zero magnetic field, so that the current noise of power supply can only bring little magnetic noise to atoms. However, the focus of the high-resolution objective, where the target layer should be located, was aligned 1 mm below the gradient center due to assembly deviation. Then, the total field strength at the position of the target layer is |B_total| = 0.1 cm × 100 G/cm + 35 G = 45 G, proportional to the current of the power supplies. Thus, the atoms suffer directly from the current noises arising from the relevant power supplies.

The radius of the atom cloud in our experiment is smaller than 50 μm. In this region, we realize a very flat magnetic field in the X–Y plane as |B_total(50 μm)| − |B_total(0)| < 0.7 mG. Meanwhile, the lattice constant in the Z direction, 532 nm, results in a Δ B = 5.32 mG (equivalent to a Zeeman shift of 2.1 × 5.32 = 11.17 kHz) difference between the neighboring layers. Therefore, equation (2) is satisfied. In principle, a magnetic field composed of a gradient field and an offset can be generated by a single coil. However, our method which combines anti-Helmholtz coils and Helmholtz coils provides the degree of freedom to modify the ratio of magnetic field gradient to field strength.

As the strong magnetic fields require large electric currents which cause dramatically heating to the coils, water cooling is employed to stabilize the temperature of these coils. Suppressing the temperature fluctuation of the coils is of great importance for two aspects: Firstly, the thermal deformation of the coils is reduced, which results in a precisely stable magnetic field around the target layer of atoms. Secondly, the thermal disturbance to the surrounding air is reduced, which results in a stable refractive index favorable for stabilizing the beam paths. Here the coils are made of hollow copper wires with the central hole radius of 1.25 mm for injecting temperature actuated water. Meanwhile, the temperatures of the coils are monitored by sensors (Texas Instruments, LM35DZ) as reference to prevent overheating.

Transferring ⁸⁷Rb atoms between |↓⟩ and |↑⟩ can be performed either by Rabi π pulses^[23] or the adiabatic LZC.^[22] However, the former is more sensitive to magnetic noises since the phase and amplitude of Rabi oscillations could be modified by the magnetic fluctuations by changing the detunings. In contrast, the latter has a much higher tolerance to magnetic noises. Therefore, we employ the LZC in our experiment for flipping the atomic sates.

The Zeeman splitting of |↑⟩ and |↓⟩ is 2.1 kHz/mG and the magnetic field gradient induces an energy difference of 11.17 kHz between neighboring layers. Thus, we perform a microwave LZC by sweeping its frequency from −4 kHz to 4 kHz with respect to the resonant frequency.

The time dependence of the microwave frequency and strength is of key importance for the transfer efficiency. To achieve a high efficiency, we use a hyperbolic secant (HS1) pulse^[24] whose Rabi frequency Ω(t) and frequency detuning Δ(t) are defined as follows:

Figure 3 illustrates the reason of using an HS1 pulse for flipping the atoms. A theoretical estimation of the transfer efficiency shows that the HS1 pulse produces a flat-top profile with the efficiency close to 1 as long as the resonant frequency is within the sweeping range, and falls sharply to nearly zero once the resonant frequency lies outside the sweeping range. This feature is particularly useful for flipping the target layer completely while preventing the atoms in the neighboring layers from being disturbed in case a moderate magnetic fluctuation is present. In contrast, a similar microwave pulse with a chirped modulation produces an undesiredly oscillating efficiency curve.

Fig. 3.

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Fig. 3. Comparison between the estimated LZC efficiencies and the measured results. Theoretical estimations with a chirped sweep (orange curve) and HS1 pulse (blue curve). Both the sweeps have the same range of ± 4 kHz around the central frequency and same pulse duration of 4 ms. For the HS1 pulse, Ωmax is set to 3.21 kHz. In chirped modulation, the Rabi frequency is fixed at Ωmax, while the microwave frequency changes with time linearly. The experimental results match very well with the theoretical estimation.

In our experiment, we measured the transfer efficiency when applying the HS1 pulse to a cloud of BEC atoms in a homogeneous magnetic field of 1 G along the Z direction. All the atoms N₀ in |↓⟩ are first transferred to |↑⟩ by a microwave LZC pulse and subsequently back to |↓⟩ by a second identical pulse. Then the number of atoms in |↑⟩ is measured as N_|↑⟩. With the knowledge of N₀ and N_|↑⟩, we can derive the transfer efficiency η given the following relations

To determine the sign in the expression of η_±, we measure the number of spin-up atoms after applying the first LZC pulse. When the measured number is larger than N_0/2, we choose +. Otherwise, we choose −. Shown in Fig. 3, our measured efficiency of the LZC process shows a good agreement with the relevant theoretical estimations. This is an evidence that the environmental magnetic noises are well suppressed in our system.

The Rabi frequency of the pulse is proportional to the square root of the microwave power. Measuring the Rabi oscillation of the atoms under a microwave power of 2 dBm delivered by the signal generator, we calibrate the Rabi frequency as 3.21(2) kHz. Under the magnetic field with a gradient of 100 G/cm, we set a sweep range of ± 800 kHz, a pulse length of 20 ms, and a Rabi frequency of 14 kHz to flip the atoms in all layers. This is called a global flip. In contrast, a range of ± 4 kHz, a pulse length of 4 ms and a Rabi frequency of 3.21 kHz are employed to flip a single layer of interest. This is called a single flip. Its efficiency is measured to be 99.92(3)%.

In principle, a shorter HS1 pulse with a higher Rabi frequency and a reduced sweep range could also be used without decreasing the efficiency under the condition of on resonance. However, with such a pulse, the two edges of the efficiency curve in Fig. 3 will be less sharp, leading to a smaller flat region of the curve in the middle of the sweep range. Therefore, for a same drift of the environmental magnetic field, the long-term stability of the efficiency gets worse compared with the one in the present experiment.

In our experiment, a typical spin-flip efficiency reaches over 99%. However, there are still about hundred atoms remained in their initial state |↓⟩ after a global flip over more than 10⁵ atoms. These atoms will create noises during final detection by fluorescence imaging. Hence, we use a σ⁻ polarized repump laser resonant with |↓⟩ → |5² P_3/2, F = 2⟩ to transfer them to |↑⟩ via a spontaneous Raman transition |↓⟩ → |5² P_3/2, F = 2 ⟩ → |↑⟩.

After the single flip pulse, only atoms in the target layer are transferred back to |↓⟩, while the rest in other layers still remain in |↑⟩. They are removed with a σ⁻ laser, the push-out laser, resonant with the circling transition |↑⟩ ← |5² P_3/2, F = 3⟩. Therefore, no dark state involves in this process resulting in a removal efficiency >99%.

Intuitively, the single-layer selection requires the noise and drift of the magnetic field to be much lower than ΔB. A numerical simulation on the LZC process shows that the magnetic fluctuation should be lower than 1 mG for maintaining a spin-flip efficiency over 99%. Since the magnetic field around the atoms is about 45 G, this corresponds to a noise level of 22 ppm. In principle, it is straightforward to detect the magnetic field with a sensor and then stabilize the field strength to a certain value by an active feedback circuit.^[25–27] In our experiment, however, the magnetic field is too strong (∼ 45 G) that no available sensor provides a proper precision at such a strength. To satisfy this stringent request, we employ three approaches: (a) Suppress the electronic noise of the current supply with a fast feedback circuit. (b) Compensate for the drift of the magnetic field induced by the drift of current supply. (c) Compensate for the drift of environmental magnetic field.

7.1. Noise reduction of the current supply

The offset coils and the gradient coils create magnetic fields at the position of atoms with the factors of 1.75 G/A and 0.1 G/A, respectively (see Fig. 2). It is important to suppress the magnetic noises generated by both coils. In our setup, an ultralow-noise commercial current supply (HighFinesse UHCS20/15) is used to power the offset coils. Its rms noise is 0.2 mA at the working current of 20 A. The current of the gradient coils is delivered by a commercial power supply (Agilent 6682 A), whose rms current noise is measured to be 5.5 mA at the current of 100 A.

We designed a noise reduction system with an insulated-gate bipolar transistor (IGBT) and a Hall sensor (LEM, ITN 600-S). The power supply, gradient coils and the IGBT are connected in series. The Hall sensor detects the current in coils and delivers a scaled current. This signal is then transformed into a voltage with a sampling resistor which has an extremely low temperature coefficient of 0.5 ppm/K. Afterwards, this voltage is fed into a PID electronic board which provides the driving signal for the gate of IGBT.

The rms noise is suppressed to 0.6 mA at the working current of 100 A as shown in Fig. 4. From the noise spectrum in Fig. 4(b), we can see that the noise reduction system has a bandwidth larger than 1 kHz. Particularly, a noise reduction of 20 dB is observed at 50 Hz. The rms noises are measured by a Hall sensor (LEM, ITN 600-S) and a digital multimeter (Agilent 34410) with a sampling rate of 10000 samples/s. Thanks to the noise reduction approach, the total rms magnetic noise induced by the noise of power supplies is estimated to be 0.36 mG.

Fig. 4.

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	Fig. 4. Comparison of the direct output from the current supply and the output from the noise reduction circuit. Current fluctuation in time domain, in which the mean value is subtracted from the original data. Current noise spectrum.

Ideally, the magnetic field gradient should be well aligned along the Z direction and orthogonal to the plane of each layers. However, in the experiment, the tilt of the coils as well as a background magnetic field may change the direction of the magnetic field gradient, resulting in a misalignment. Suppose a background magnetic field along the X direction (B = Be_x), an angle θ will be created between the direction of gradient and the Z axis as

Fig. 5.

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	Fig. 5. in situ imaging of the atoms selected from a few layers of the 1D optical lattice. (a) Blurred fringes due to the noises of magnetic field before applying the noise reduction system. (b) The fringes get much clear after the noise reduction system is optimized.

As shown in Fig. 5, we applied an offset magnetic field on the X–Y plane using the compensation coils to tilt the gradient. Before the noises of the magnetic fields are suppressed, we only observe blurred fringes as in Fig. 5(a), indicating a low LZC efficiency. In contrast, the fringes get clear when we implement the noise reduction system, showing a high LZC efficiency.

In order to study the drift of magnetic field around the atoms, we run the above experiment repeatedly to monitor the drift of fringes. An 8-hour measurement was performed, as shown in Fig. 6. The period, 12.4(4) μm, of the fringes indicates a tilt angle of θ = 2.46(8)°. The rms width of the curve in Fig. 6 shows a short-term drift of the magnetic field within the range of 0.23 mG. Meanwhile, by fitting the data of fringe 2 with a linear function, a clear long-term drift (in red, lower panel) with an average drift speed of 0.042(3) mG/hour is obtained. The sweep range of the HS1 pulse for selecting one layer is ± 4 kHz, corresponding to a range of magnetic field of ± 1.9 mG. Therefore, preparing the 2D single layer of atoms can be continuously carried out with a high efficiency for 48 hours before the resonant condition is gone. This means that we only need to re-adjust the frequency center once a day.

Fig. 6.

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Fig. 6. Characterization of the drift of magnetic field. (a) Drift of center of the stripes indicates the drift of magnetic field. (b) Drift of the magnetic field under different temperature fluctuations of the coils. To obtain the center of stripes, we integrates the counts in each row to get the curve and then fits it with a sum of gaussian functions to get the position of peaks, as shown in right panels of Fig. 5. The error bars show 1σ statistical uncertainty of the fitted centers. The relationship between the drift of stripes and the drift of magnetic fields is estimated with the magnetic field gradient and the structure of stripes. (c) An exemplary image of the atoms selected in a single layer when the noise and drift of the magnetic fields are suppressed.

Besides the three approaches introduced in Section 7, we find that the stability of the generated magnetic field suffers from the temperature fluctuation of the coils due to thermal deformations which should be suppressed as well. Figure 6(b) shows the two measurements on the phase of the fringes at different temperature fluctuations. The orange data points were taken when there was a temperature oscillation with an amplitude of ± 0.7 K. A period of 3.3 min is observed in the drift of the strength of magnetic field, which matches the temperature change of the coils and of the cooling water delivered by the chiller. To minimize this effect, we employed a better water chiller for coils and the temperature fluctuation was decreased to ± 0.1 K. As a result, the oscillation amplitude of the magnetic field is reduced from ± 0.75 mG to ± 0.25 mG (blue data points).

It is worth mentioning that some groups synchronize their experimental sequence to the AC line phase to reduce the influence of 50 Hz noise.^[21,28] However, according to our test, it is unnecessary to do so since the above approaches suppress such a noise effectively. After the misalignment of magnetic field gradient is corrected, we always achieve an efficiency higher than 99% for preparing the single layer which does not depend on the synchronization. Figure 6(c) shows an exemplary image of the atoms selected in a single layer shoot with the microscope when the noise and drift of the magnetic fields are suppressed.

We have developed a technique on preparing a 2D quantum gas based on slicing a single layer of atoms trapped in a few layers of pancake-shaped optical traps. To achieve a high efficiency for this preparation, a Landau–Zener-cross microwave pulse with well-designed profile is applied to improve the noise tolerance and stability. The rms noise of magnetic field induced by the current supplies is reduced to 0.36 mG with a home-built noise reduction system. The slow drift of magnetic field is minimized with a static compensation circuit and a more stable water chiller. The effectiveness of these approaches was verified by observing the response of atoms with in situ microscopic imaging. Finally a slow drift speed of 0.042(3) mG/hour is achieved, promising a continuous experiment of 48 hours for maintaining the preparation efficiency of the 2D quantum gas as 99.92(3)%. Synchronizing the experimental sequence to the phase of the AC line is found to be unnecessary for achieving the high efficiency.

Our work can be extended to the creation of a bilayer lattice system, in which disentangling two layers to create a low-entropy single layer system is proposed.^[29] Meanwhile, the home-made ultrastable current supply and the full scheme of stabilizing magnetic fields have promising applications in both scientific researches and various industries.